Munich Chain Ladder Closing the gap between paid and incurred IBNR estimates Dr. Gerhard Quarg

1. Münchener RückMunich Re Group Munich Chain Ladder Closing the gap between paid and incurred IBNR estimates Dr. Gerhard Quarg

2. From Chain Ladder to MunichChainLadder

3. Triangle of P/I ratios vs. development years

4. P/I quadrangle (with separate Chain Ladder estimates)

5. Applying Chain Ladder to paid and incurred separately • Interpretation of graphic: A low current P/I ratio yields a low projected ultimate P/I ratio. A high current P/I ratio yields a high projected ultimate P/I ratio. • This is inherent to separate Chain Ladder calculations. Thorough mathematical formulation: For each accident year, the quotient of the ultimate P/I ratio and the average ultimate P/I ratio and the quotient of the current P/I ratio and the corresponding average P/I ratio agree.

6. Correlations between paid and incurred data • Explanation: Below-average P/I ratios were succeeded by relatively high paid and/or relatively low incurred development factors. Above-average P/I ratios were succeeded by relatively low paid and/or relatively high incurred development factors. • This can indeed be seen in the data:

7. Paid development factors vs. preceding P/I ratios

8. Incurred development factors vs. preceding P/I ratios

9. Problem: paid dev. factors vs. widely scattered P/I ratios

10. Solution (Th. Mack): paid dev. factors vs. I/P ratios

11. The residual approach • Problem: high volatility due to not enough data, especially in later development years • Solution: consider all development years together Use residuals to make different development years comparable. Residuals measure deviations from the expected value in multiples of the standard deviation.

12. Residuals of paid development factors vs. I/P residuals

13. Residuals of incurred dev. factors vs. P/I residuals

14. The stochastic MCL model

15. where and The standard Chain Ladder model • Main assumptions of the standard Chain Ladder model: • These assumptions are designed for the projection of one triangle. • They ignore systematic correlations between paid losses and incurred losses.

16. where Res ( ) denotes the conditional residual. Required model features In order to combine paid and incurred information we need or equivalently

17. Lambda is the slope of the regression line through the origin in the respective residual plot. The new model: Munich Chain Ladder • The Munich Chain Ladder assumptions:

18. The new model: Munich Chain Ladder • Interpretation of lambda as correlation parameter: • Together, both lambda parameters characterise the interdependency of paid and incurred.

19. The new model: Munich Chain Ladder • The Munich Chain Ladder recursion formulas: • Lambda is the slope of the regression line through the origin in the residual plot, sigma and rho are variance parameters and q is the average P/I ratio.

20. Capability and limits of Munich Chain Ladder

21. Initial example: ult. P/I ratios (separate CL vs. MCL)

22. Triangle of P/I ratios vs. development years

23. P/I quadrangle (with separate Chain Ladder estimates)

24. P/I quadrangle (with Munich Chain Ladder)

25. Another example: ultimate P/I ratios (SCL vs. MCL)

26. The remaining gap • Munich Chain Ladder projects ultimate P/I ratios of about “only” 96%. • There is a remaining gap between paid and incurred IBNR estimates. • This is not a failure of Munich Chain Ladder. On the contrary, data suggest a remaining gap after 14 years of development:

27. Triangle of P/I ratios vs. development years

28. Münchener RückMunich Re Group Thank you for your interest Dr. Gerhard Quarg